Also, a 1960 Nixon Presidency would create several major butterflies. It is likely that the war against Castro that was in the works in the later Eisenhower administration would go forward, but with a greater direct US military involvement from the start. This means (probably) no Bay of Pigs fiasco, but much greater tensions with the USSR. Khruschev was not about to go to war with the US over a minor and fairly new client, but would certainly find some way to strike back through a proxy, possibly in the Middle East or Latin America. There would be no Cuban Missile Crisis, but there might be an ongoing guerrilla war in the hills of Cuba for some time to come.
Second, Nixon might have handled Viet Nam differently post 1964. Ike and MacArthur would both have advised against a massive intervention, but the war might have developed as IOTL. Certainly Nixon proved in 1972 that he was quite willing to take steps against the North that LBJ never did. So Nixon might have had a couple of political-military issues that would have reduced his popularity in a second term, but probably would have had a fairly successful first.
As for Democratic nominees in 1964, maybe Stu Symington or Pat Brown.